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Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. (English) Zbl 1317.60007

The authors study a general framework for regular variation and the so-called heavy tails for distributions on metric-space-valued random variables, with further applications to regular variation for measures on \(\mathbb{R}_+= [0,+\infty)\) and on the space of all real-valued, right-continuous functions having left limits on \([0,1]\).
The heavy tails occur in a variety of domains such as risk management, quantitative finance and economics, complex networks of data and telecommunication transmissions, and social networks. In the particular case of one dimension, the heavy tails are known as Pareto tails. The general mathematical framework for developing the heavy tails is the theory of regular variation, originally formulated on \([0,+\infty)\) and extended to more general spaces; see [N. H. Bingham et al., Regular variation. Cambridge University Press (1987; Zbl 0617.26001); Paperback ed. (1989; Zbl 0667.26003)].
Finally, the authors attempt to clarify the proper definition of regular variation in metric spaces.

MSC:

60B05 Probability measures on topological spaces
28A33 Spaces of measures, convergence of measures
60G17 Sample path properties
60G51 Processes with independent increments; Lévy processes
60G70 Extreme value theory; extremal stochastic processes
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References:

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